The generalized bounded-potential conjecture for nef (1,1)-classes

Let $X$ be a compact Kähler manifold with $K_X$ pseudoeffective, and let $[\alpha]$ be a nef $(1,1)$-class such that

$$\lambda[\alpha]-c_1(K_X)$$

is nef for some $\lambda\in\mathbb{R}_{>0}$. Generalized bounded-potential conjecture. Then $[\alpha]$ contains a closed positive current with bounded potentials. This generalization is presented as an analogue of the generalized abundance conjecture and remains unresolved.